A 2-category is by definition a category enriched over categories. Hence, there is a default notion of morphisms between 2-categories, called 2-functors, which are strict functors that respect the categorical enrichment. On the other hand, we can also consider pseudofunctors, for which the composition and unit hold only up to chosen natural isomorphisms. For a more detailed discussion of pseudofunctors see [1]}, [2]}.
